Random Numbers part 9

Random Numbers part 9

Adaptive and Recursive Monte Carlo Methods
This section discusses more advanced techniques of Monte Carlo integration. As examples of the use of these techniques, we include two rather different, fairly sophisticated, multidimensional Monte Carlo codes: vegas [1,2] , and miser [3]. The techniques that we discuss all fall under the general rubric of reduction of variance (§7.6), but are otherwise quite distinct.

Nội dung Text: Random Numbers part 9

316 Chapter 7. Random Numbers
7.8 Adaptive and Recursive Monte Carlo
Methods
This section discusses more advanced techniques of Monte Carlo integration. As
examples of the use of these techniques, we include two rather different, fairly sophisticated,
multidimensional Monte Carlo codes: vegas [1,2] , and miser [3]. The techniques that we
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discuss all fall under the general rubric of reduction of variance (§7.6), but are otherwise
quite distinct.
Importance Sampling
The use of importance sampling was already implicit in equations (7.6.6) and (7.6.7).
We now return to it in a slightly more formal way. Suppose that an integrand f can be written
as the product of a function h that is almost constant times another, positive, function g. Then
its integral over a multidimensional volume V is
f dV = (f /g) gdV = h gdV (7.8.1)
In equation (7.6.7) we interpreted equation (7.8.1) as suggesting a change of variable to
G, the indeﬁnite integral of g. That made gdV a perfect differential. We then proceeded
to use the basic theorem of Monte Carlo integration, equation (7.6.1). A more general
interpretation of equation (7.8.1) is that we can integrate f by instead sampling h — not,
however, with uniform probability density dV , but rather with nonuniform density gdV . In
this second interpretation, the ﬁrst interpretation follows as the special case, where the means
of generating the nonuniform sampling of gdV is via the transformation method, using the
indeﬁnite integral G (see §7.2).
More directly, one can go back and generalize the basic theorem (7.6.1) to the case
of nonuniform sampling: Suppose that points xi are chosen within the volume V with a
probability density p satisfying
p dV = 1 (7.8.2)
The generalized fundamental theorem is that the integral of any function f is estimated, using
N sample points xi , . . . , xN , by
f f f 2 /p2 − f /p 2
I≡ f dV = pdV ≈ ± (7.8.3)
p p N
where angle brackets denote arithmetic means over the N points, exactly as in equation
(7.6.2). As in equation (7.6.1), the “plus-or-minus” term is a one standard deviation error
estimate. Notice that equation (7.6.1) is in fact the special case of equation (7.8.3), with
p = constant = 1/V .
What is the best choice for the sampling density p? Intuitively, we have already
seen that the idea is to make h = f /p as close to constant as possible. We can be more
rigorous by focusing on the numerator inside the square root in equation (7.8.3), which is
the variance per sample point. Both angle brackets are themselves Monte Carlo estimators
of integrals, so we can write
2 2 2
f2 f f2 f f2
S≡ − ≈ pdV − pdV = dV − f dV (7.8.4)
p2 p p2 p p
We now ﬁnd the optimal p subject to the constraint equation (7.8.2) by the functional variation
2
δ f2
0= dV − f dV +λ p dV (7.8.5)
δp p

7.8 Adaptive and Recursive Monte Carlo Methods 317
with λ a Lagrange multiplier. Note that the middle term does not depend on p. The variation
(which comes inside the integrals) gives 0 = −f 2 /p2 + λ or
|f | |f |
p= √ = (7.8.6)
λ |f | dV
where λ has been chosen to enforce the constraint (7.8.2).
If f has one sign in the region of integration, then we get the obvious result that the
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optimal choice of p — if one can ﬁgure out a practical way of effecting the sampling — is
that it be proportional to |f |. Then the variance is reduced to zero. Not so obvious, but seen
to be true, is the fact that p ∝ |f | is optimal even if f takes on both signs. In that case the
variance per sample point (from equations 7.8.4 and 7.8.6) is
2 2
S = Soptimal = |f | dV − f dV (7.8.7)
One curiosity is that one can add a constant to the integrand to make it all of one sign,
since this changes the integral by a known amount, constant × V . Then, the optimal choice
of p always gives zero variance, that is, a perfectly accurate integral! The resolution of
this seeming paradox (already mentioned at the end of §7.6) is that perfect knowledge of p
in equation (7.8.6) requires perfect knowledge of |f |dV , which is tantamount to already
knowing the integral you are trying to compute!
If your function f takes on a known constant value in most of the volume V , it is
certainly a good idea to add a constant so as to make that value zero. Having done that, the
accuracy attainable by importance sampling depends in practice not on how small equation
(7.8.7) is, but rather on how small is equation (7.8.4) for an implementable p, likely only a
crude approximation to the ideal.
Stratiﬁed Sampling
The idea of stratiﬁed sampling is quite different from importance sampling. Let us
expand our notation slightly and let f denote the true average of the function f over
the volume V (namely the integral divided by V ), while f denotes as before the simplest
(uniformly sampled) Monte Carlo estimator of that average:
1 1
f ≡ f dV f ≡ f (xi) (7.8.8)
V N i
The variance of the estimator, Var ( f ), which measures the square of the error of the
Monte Carlo integration, is asymptotically related to the variance of the function, Var (f ) ≡
f 2 − f 2, by the relation
Var (f )
Var ( f ) = (7.8.9)
N
(compare equation 7.6.1).
Suppose we divide the volume V into two equal, disjoint subvolumes, denoted a and b,
and sample N/2 points in each subvolume. Then another estimator for f , different from
equation (7.8.8), which we denote f , is
1
f ≡
f a+ f b (7.8.10)
2
in other words, the mean of the sample averages in the two half-regions. The variance of
estimator (7.8.10) is given by
1
Var f = Var f a + Var f b
4
1 Vara (f ) Varb (f )
= + (7.8.11)
4 N/2 N/2
1
= [Vara (f ) + Varb (f )]
2N

318 Chapter 7. Random Numbers
Here Vara (f ) denotes the variance of f in subregion a, that is, f 2 a − f 2
a, and
correspondingly for b.
From the deﬁnitions already given, it is not difﬁcult to prove the relation
1 1
Var (f ) = [Vara (f ) + Varb (f )] + ( f a − f b)2 (7.8.12)
2 4
(In physics, this formula for combining second moments is the “parallel axis theorem.”)
Comparing equations (7.8.9), (7.8.11), and (7.8.12), one sees that the stratiﬁed (into two
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subvolumes) sampling gives a variance that is never larger than the simple Monte Carlo case
— and smaller whenever the means of the stratiﬁed samples, f a and f b, are different.
We have not yet exploited the possibility of sampling the two subvolumes with different
numbers of points, say Na in subregion a and Nb ≡ N − Na in subregion b. Let us do so
now. Then the variance of the estimator is
1 Vara (f ) Varb (f )
Var f = + (7.8.13)
4 Na N − Na
which is minimized (one can easily verify) when
Na σa
= (7.8.14)
N σa + σb
Here we have adopted the shorthand notation σa ≡ [Vara (f )]1/2 , and correspondingly for b.
If Na satisﬁes equation (7.8.14), then equation (7.8.13) reduces to
(σa + σb )2
Var f = (7.8.15)
4N
Equation (7.8.15) reduces to equation (7.8.9) if Var (f ) = Vara (f ) = Varb (f ), in which case
stratifying the sample makes no difference.
A standard way to generalize the above result is to consider the volume V divided into
more than two equal subregions. One can readily obtain the result that the optimal allocation of
sample points among the regions is to have the number of points in each region j proportional
to σj (that is, the square root of the variance of the function f in that subregion). In spaces
of high dimensionality (say d > 4) this is not in practice very useful, however. Dividing a
∼
volume into K segments along each dimension implies K d subvolumes, typically much too
large a number when one contemplates estimating all the corresponding σj ’s.
Mixed Strategies
Importance sampling and stratiﬁed sampling seem, at ﬁrst sight, inconsistent with each
other. The former concentrates sample points where the magnitude of the integrand |f | is
largest, that latter where the variance of f is largest. How can both be right?
The answer is that (like so much else in life) it all depends on what you know and how
well you know it. Importance sampling depends on already knowing some approximation to
your integral, so that you are able to generate random points xi with the desired probability
density p. To the extent that your p is not ideal, you are left with an error that decreases
only as N −1/2 . Things are particularly bad if your p is far from ideal in a region where the
integrand f is changing rapidly, since then the sampled function h = f /p will have a large
variance. Importance sampling works by smoothing the values of the sampled function h,
and is effective only to the extent that you succeed in this.
Stratiﬁed sampling, by contrast, does not necessarily require that you know anything
about f . Stratiﬁed sampling works by smoothing out the ﬂuctuations of the number of points
in subregions, not by smoothing the values of the points. The simplest stratiﬁed strategy,
dividing V into N equal subregions and choosing one point randomly in each subregion,
already gives a method whose error decreases asymptotically as N −1 , much faster than
N −1/2 . (Note that quasi-random numbers, §7.7, are another way of smoothing ﬂuctuations in
the density of points, giving nearly as good a result as the “blind” stratiﬁcation strategy.)
However, “asymptotically” is an important caveat: For example, if the integrand is
negligible in all but a single subregion, then the resulting one-sample integration is all but

7.8 Adaptive and Recursive Monte Carlo Methods 319
useless. Information, even very crude, allowing importance sampling to put many points in
the active subregion would be much better than blind stratiﬁed sampling.
Stratiﬁed sampling really comes into its own if you have some way of estimating the
variances, so that you can put unequal numbers of points in different subregions, according to
(7.8.14) or its generalizations, and if you can ﬁnd a way of dividing a region into a practical
number of subregions (notably not K d with large dimension d), while yet signiﬁcantly
reducing the variance of the function in each subregion compared to its variance in the full
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volume. Doing this requires a lot of knowledge about f , though different knowledge from
what is required for importance sampling.
In practice, importance sampling and stratiﬁed sampling are not incompatible. In many,
if not most, cases of interest, the integrand f is small everywhere in V except for a small
fractional volume of “active regions.” In these regions the magnitude of |f | and the standard
deviation σ = [Var (f )]1/2 are comparable in size, so both techniques will give about the
same concentration of points. In more sophisticated implementations, it is also possible to
“nest” the two techniques, so that (e.g.) importance sampling on a crude grid is followed
by stratiﬁcation within each grid cell.
Adaptive Monte Carlo: VEGAS
The VEGAS algorithm, invented by Peter Lepage [1,2] , is widely used for multidimen-
sional integrals that occur in elementary particle physics. VEGAS is primarily based on
importance sampling, but it also does some stratiﬁed sampling if the dimension d is small
enough to avoid K d explosion (speciﬁcally, if (K/2)d < N/2, with N the number of sample
points). The basic technique for importance sampling in VEGAS is to construct, adaptively,
a multidimensional weight function g that is separable,
p ∝ g(x, y, z, . . .) = gx (x)gy (y)gz (z) . . . (7.8.16)
Such a function avoids the K d explosion in two ways: (i) It can be stored in the computer
as d separate one-dimensional functions, each deﬁned by K tabulated values, say — so that
K × d replaces K d. (ii) It can be sampled as a probability density by consecutively sampling
the d one-dimensional functions to obtain coordinate vector components (x, y, z, . . .).
The optimal separable weight function can be shown to be [1]
1/2
f 2 (x, y, z, . . .)
gx (x) ∝ dy dz . . . (7.8.17)
gy (y)gz (z) . . .
(and correspondingly for y, z, . . .). Notice that this reduces to g ∝ |f | (7.8.6) in one
dimension. Equation (7.8.17) immediately suggests VEGAS’ adaptive strategy: Given a
set of g-functions (initially all constant, say), one samples the function f , accumulating not
only the overall estimator of the integral, but also the Kd estimators (K subdivisions of the
independent variable in each of d dimensions) of the right-hand side of equation (7.8.17).
These then determine improved g functions for the next iteration.
When the integrand f is concentrated in one, or at most a few, regions in d-space, then
the weight function g’s quickly become large at coordinate values that are the projections of
these regions onto the coordinate axes. The accuracy of the Monte Carlo integration is then
enormously enhanced over what simple Monte Carlo would give.
The weakness of VEGAS is the obvious one: To the extent that the projection of the
function f onto individual coordinate directions is uniform, VEGAS gives no concentration
of sample points in those dimensions. The worst case for VEGAS, e.g., is an integrand that
is concentrated close to a body diagonal line, e.g., one from (0, 0, 0, . . .) to (1, 1, 1, . . .).
Since this geometry is completely nonseparable, VEGAS can give no advantage at all. More
generally, VEGAS may not do well when the integrand is concentrated in one-dimensional
(or higher) curved trajectories (or hypersurfaces), unless these happen to be oriented close
to the coordinate directions.
The routine vegas that follows is essentially Lepage’s standard version, minimally
modiﬁed to conform to our conventions. (We thank Lepage for permission to reproduce the
program here.) For consistency with other versions of the VEGAS algorithm in circulation,

320 Chapter 7. Random Numbers
we have preserved original variable names. The parameter NDMX is what we have called K,
the maximum number of increments along each axis; MXDIM is the maximum value of d; some
other parameters are explained in the comments.
The vegas routine performs m = itmx statistically independent evaluations of the
desired integral, each with N = ncall function evaluations. While statistically independent,
these iterations do assist each other, since each one is used to reﬁne the sampling grid for
the next one. The results of all iterations are combined into a single best answer, and its
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estimated error, by the relations
m m m −1/2
Ii 1 1
Ibest = 2 2
σbest = 2
(7.8.18)
i=1
σi i=1
σi i=1
σi
Also returned is the quantity
m
1 (Ii − Ibest)2
χ2 /m ≡ (7.8.19)
m−1 i=1
σi2
If this is signiﬁcantly larger than 1, then the results of the iterations are statistically
inconsistent, and the answers are suspect.
The input ﬂag init can be used to advantage. One might have a call with init=0,
ncall=1000, itmx=5 immediately followed by a call with init=1, ncall=100000, itmx=1.
The effect would be to develop a sampling grid over 5 iterations of a small number of samples,
then to do a single high accuracy integration on the optimized grid.
Note that the user-supplied integrand function, fxn, has an argument wgt in addition
to the expected evaluation point x. In most applications you ignore wgt inside the function.
Occasionally, however, you may want to integrate some additional function or functions along
with the principal function f . The integral of any such function g can be estimated by
Ig = wi g(x) (7.8.20)
i
where the wi ’s and x’s are the arguments wgt and x, respectively. It is straightforward to
accumulate this sum inside your function fxn, and to pass the answer back to your main
program via global variables. Of course, g(x) had better resemble the principal function f to
some degree, since the sampling will be optimized for f .
#include
#include
#include "nrutil.h"
#define ALPH 1.5
#define NDMX 50
#define MXDIM 10
#define TINY 1.0e-30
extern long idum; For random number initialization in main.
void vegas(float regn[], int ndim, float (*fxn)(float [], float), int init,
unsigned long ncall, int itmx, int nprn, float *tgral, float *sd,
float *chi2a)
Performs Monte Carlo integration of a user-supplied ndim-dimensional function fxn over a
rectangular volume speciﬁed by regn[1..2*ndim], a vector consisting of ndim “lower left”
coordinates of the region followed by ndim “upper right” coordinates. The integration consists
of itmx iterations, each with approximately ncall calls to the function. After each iteration
the grid is reﬁned; more than 5 or 10 iterations are rarely useful. The input ﬂag init signals
whether this call is a new start, or a subsequent call for additional iterations (see comments
below). The input ﬂag nprn (normally 0) controls the amount of diagnostic output. Returned
answers are tgral (the best estimate of the integral), sd (its standard deviation), and chi2a
(χ2 per degree of freedom, an indicator of whether consistent results are being obtained). See
text for further details.
{
float ran2(long *idum);

324 Chapter 7. Random Numbers
The starting points are equations (7.8.10) and (7.8.13), applied to bisections of successively
smaller subregions.
Suppose that we have a quota of N evaluations of the function f , and want to evaluate
f in the rectangular parallelepiped region R = (xa , xb ). (We denote such a region by the
two coordinate vectors of its diagonally opposite corners.) First, we allocate a fraction p of
N towards exploring the variance of f in R: We sample pN function values uniformly in
R and accumulate the sums that will give the d different pairs of variances corresponding to
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the d different coordinate directions along which R can be bisected. In other words, in pN
samples, we estimate Var (f ) in each of the regions resulting from a possible bisection of R,
1
Rai ≡(xa , xb − ei · (xb − xa )ei )
2 (7.8.21)
1
Rbi ≡(xa + ei · (xb − xa )ei , xb )
2
Here ei is the unit vector in the ith coordinate direction, i = 1, 2, . . . , d.
Second, we inspect the variances to ﬁnd the most favorable dimension i to bisect. By
equation (7.8.15), we could, for example, choose that i for which the sum of the square roots
of the variance estimators in regions Rai and Rbi is minimized. (Actually, as we will explain,
we do something slightly different.)
Third, we allocate the remaining (1 − p)N function evaluations between the regions
Rai and Rbi . If we used equation (7.8.15) to choose i, we should do this allocation according
to equation (7.8.14).
We now have two parallelepipeds each with its own allocation of function evaluations
for estimating the mean of f . Our “RSS” algorithm now shows itself to be recursive: To
evaluate the mean in each region, we go back to the sentence beginning “First,...” in the
paragraph above equation (7.8.21). (Of course, when the allocation of points to a region falls
below some number, we resort to simple Monte Carlo rather than continue with the recursion.)
Finally, we combine the means, and also estimated variances of the two subvolumes,
using equation (7.8.10) and the ﬁrst line of equation (7.8.11).
This completes the RSS algorithm in its simplest form. Before we describe some
additional tricks under the general rubric of “implementation details,” we need to return
brieﬂy to equations (7.8.13)–(7.8.15) and derive the equations that we actually use instead of
these. The right-hand side of equation (7.8.13) applies the familiar scaling law of equation
(7.8.9) twice, once to a and again to b. This would be correct if the estimates f a and f b
were each made by simple Monte Carlo, with uniformly random sample points. However, the
two estimates of the mean are in fact made recursively. Thus, there is no reason to expect
equation (7.8.9) to hold. Rather, we might substitute for equation (7.8.13) the relation,
1 Vara (f ) Varb (f )
Var f = + (7.8.22)
4 Naα (N − Na )α
where α is an unknown constant ≥ 1 (the case of equality corresponding to simple Monte
Carlo). In that case, a short calculation shows that Var f is minimized when
Na Vara (f )1/(1+α)
= (7.8.23)
N Vara (f )1/(1+α) + Varb (f )1/(1+α)
and that its minimum value is
1+α
Var f ∝ Vara (f )1/(1+α) + Varb (f )1/(1+α) (7.8.24)
Equations (7.8.22)–(7.8.24) reduce to equations (7.8.13)–(7.8.15) when α = 1. Numerical
experiments to ﬁnd a self-consistent value for α ﬁnd that α ≈ 2. That is, when equation
(7.8.23) with α = 2 is used recursively to allocate sample opportunities, the observed variance
of the RSS algorithm goes approximately as N −2 , while any other value of α in equation
(7.8.23) gives a poorer fall-off. (The sensitivity to α is, however, not very great; it is not
known whether α = 2 is an analytically justiﬁable result, or only a useful heuristic.)
The principal difference between miser’s implementation and the algorithm as described
thus far lies in how the variances on the right-hand side of equation (7.8.23) are estimated.

7.8 Adaptive and Recursive Monte Carlo Methods 325
We ﬁnd empirically that it is somewhat more robust to use the square of the difference of
maximum and minimum sampled function values, instead of the genuine second moment
of the samples. This estimator is of course increasingly biased with increasing sample
size; however, equation (7.8.23) uses it only to compare two subvolumes (a and b) having
approximately equal numbers of samples. The “max minus min” estimator proves its worth
when the preliminary sampling yields only a single point, or small number of points, in active
regions of the integrand. In many realistic cases, these are indicators of nearby regions of
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even greater importance, and it is useful to let them attract the greater sampling weight that
“max minus min” provides.
A second modiﬁcation embodied in the code is the introduction of a “dithering parameter,”
dith, whose nonzero value causes subvolumes to be divided not exactly down the middle, but
rather into fractions 0.5±dith, with the sign of the ± randomly chosen by a built-in random
number routine. Normally dith can be set to zero. However, there is a large advantage in
taking dith to be nonzero if some special symmetry of the integrand puts the active region
exactly at the midpoint of the region, or at the center of some power-of-two submultiple of
the region. One wants to avoid the extreme case of the active region being evenly divided
into 2d abutting corners of a d-dimensional space. A typical nonzero value of dith, on
those occasions when it is useful, might be 0.1. Of course, when the dithering parameter
is nonzero, we must take the differing sizes of the subvolumes into account; the code does
this through the variable fracl.
One ﬁnal feature in the code deserves mention. The RSS algorithm uses a single set
of sample points to evaluate equation (7.8.23) in all d directions. At bottom levels of the
recursion, the number of sample points can be quite small. Although rare, it can happen that
in one direction all the samples are in one half of the volume; in that case, that direction
is ignored as a candidate for bifurcation. Even more rare is the possibility that all of the
samples are in one half of the volume in all directions. In this case, a random direction is
chosen. If this happens too often in your application, then you should increase MNPT (see
line if (!jb). . . in the code).
Note that miser, as given, returns as ave an estimate of the average function value
f , not the integral of f over the region. The routine vegas, adopting the other convention,
returns as tgral the integral. The two conventions are of course trivially related, by equation
(7.8.8), since the volume V of the rectangular region is known.
#include
#include
#include "nrutil.h"
#define PFAC 0.1
#define MNPT 15
#define MNBS 60
#define TINY 1.0e-30
#define BIG 1.0e30
Here PFAC is the fraction of remaining function evaluations used at each stage to explore the
variance of func. At least MNPT function evaluations are performed in any terminal subregion;
a subregion is further bisected only if at least MNBS function evaluations are available. We take
MNBS = 4*MNPT.
static long iran=0;
void miser(float (*func)(float []), float regn[], int ndim, unsigned long npts,
float dith, float *ave, float *var)
Monte Carlo samples a user-supplied ndim-dimensional function func in a rectangular volume
speciﬁed by regn[1..2*ndim], a vector consisting of ndim “lower-left” coordinates of the
region followed by ndim “upper-right” coordinates. The function is sampled a total of npts
times, at locations determined by the method of recursive stratiﬁed sampling. The mean value
of the function in the region is returned as ave; an estimate of the statistical uncertainty of ave
(square of standard deviation) is returned as var. The input parameter dith should normally
be set to zero, but can be set to (e.g.) 0.1 if func’s active region falls on the boundary of a
power-of-two subdivision of region.
{
void ranpt(float pt[], float regn[], int n);